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stmt 1 is sufficient, because the product will always be = 0, unique answer. so suff

in stmt 2, the 2 smallest integers can be -2 and -4, and the other two will be 0 and 2. In this case, product is 0, not positive.they could also be -8 and -6, then other two integers can be -4, -2 and then product will be positive. No unique answer, so insuff.

Now it also says that their sum is positive but less than 20. Sum can only be positive if the 4 numbers have atleast few numbers that are positive, ie, all four cannot be negative. So these numbers must include zero as well, (zero is considered as an even integer)

Now it also says that their sum is positive but less than 20. Sum can only be positive if the 4 numbers have atleast few numbers that are positive, ie, all four cannot be negative. So these numbers must include zero as well, (zero is considered as an even integer)

If zero is one of them, product will always be zero

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BUT: in DS statements never contradict. Which means that BOTH statement must be true. Now, from (1) we got that only possible sets are {-2,0,2,4} and {0,2,4,6}. If we take the statement (2) the product of the smallest two terms must be positive, but in these sets the product of smallest terms equals to zero (-2*0=0 and 0*2=0), which is not positive. Statements contradict.

If I'm not wrong in above, I'd suggest to change the statement (2) as follows:

(2) The product of the smallest two of these integers is not positive.

In this case answer would be D.

OR

(2) The product of the middle two of these integers is positive.

In this case answer still would be A. As it's possible to have {0,2,4,6} --> product zero OR {2,4,6,8} --> product positive OR {-8,-6,-4,-2}--> product positive. _________________